Sorry for the poor title I am really struggling to put the question into words. There is a proof that I am trying to follow but I don't understand the last step. This is the proof:
Firstly I don't follow how $p_M(x) = \det(A-xI_n)\det(B-xI_m)$
Furthermore, does this not imply that $p_M(x) = \det((A-xI_n)(B-xI_m))$ but $A$ and $B$ are both square matrices of potentially different sizes in which case $AB$ wouldn't exist?